$Z_2$ Topological Order and the Quantum Spin Hall Effect

TL;DR

The Z2 order of the QSH phase is established in the two band model of graphene and a generalization of the formalism applicable to multiband and interacting systems is proposed.

Abstract

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel $Z_2$ topological invariant, which distinguishes it from an ordinary insulator. The $Z_2$ classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the $Z_2$ order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multi band and interacting systems.

Figures (2)

• Figure 1: Energy bands for a one dimensional "zigzag" strip in the (a) QSH phase $\lambda_v=.1 t$ and (b) the insulating phase $\lambda_v=.4 t$. In both cases $\lambda_{SO}=.06 t$ and $\lambda_R=.05 t$. The edge states on a given edge cross at $ka = \pi$. The inset shows the phase diagram as a function of $\lambda_v$ and $\lambda_R$ for $0<\lambda_{SO}\ll t$.
• Figure 2: The zeros of $P({\bf k})$ in the QSH phase occur at points $\pm{\bf k}^*$ for (a) $\lambda_v\ne 0$ and on the oval for (b) $\lambda_v=0$. (c) $|P(0,\alpha_2)|$ in the QSH (solid) and insulating (dashed) phases for a $2\times 2$ supercell using parameters in Fig. 1. (d) Point ($\lambda_v\ne 0$) and line($\lambda_v=0$) zeros of $P({\vec{\alpha}})$ for the $2\times 2$ supercell. In (a,b,d) the solid dots are TR symmetric points, which can't be zeros of $P$, and $C$ is the contour of integration for Eq. (5).